Signal Processing Methods for Steering to an Underground Target

ABSTRACT

A method of processing data from an electromagnetic resistivity logging tool which includes a transmitter coil and a receiver coil is disclosed. The electromagnetic resistivity logging tool is placed at a desired location. The transmitter coil and the receiver coil are positioned at a first azimuthal angle. A signal is transmitted from the receiver coil. The receiver coil then receives a signal. The signal at the receiver coil, a tilt angle of the transmitter coil, a tilt angle of the receiver coil and the first azimuthal angle are then used to calculate a first complex voltage representing at least one component of the received signal.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to and is a divisional of U.S. patentapplication Ser. No. 13/855,408 , filed on Apr. 2, 2013, and entitled“Signal Processing Methods for Steering to an Underground Target,” whichis a continuation of International Application No. PCT/US2011/027353,filed Mar. 7, 2011, the entire disclosures of which are incorporatedherein by reference.

BACKGROUND

The basic techniques for electromagnetic logging for earth formationsare well known. For instance, induction logging to determine resistivity(or its inverse, conductivity) of earth formations adjacent a boreholehas long been a standard and important technique in the search for andrecovery of hydrocarbons. Generally, a transmitter transmits anelectromagnetic signal that passes through formation materials aroundthe borehole and induces a signal in one or more receivers. Theproperties of the signal received, such as its amplitude and/or phase,are influenced by the formation resistivity, enabling resistivitymeasurements to be made. The measured signal characteristics and/orformation properties calculated therefrom may be recorded as a functionof the tool's depth or position in the borehole, yielding a formationlog that can be used to analyze the formation.

The resistivity of a given formation may be isotropic (equal in alldirections) or anisotropic (unequal in different directions). Inelectrically anisotropic formations, the anisotropy is generallyattributable to extremely fine layering during the sedimentary build-upof the formation. As a result, in a formation Cartesian coordinatesystem oriented such that the x-y plane is parallel to the formationlayers and the z axis is perpendicular to the formation layers,resistivities Rx and Ry in the x and y directions, respectively, tend tobe similar, but resistivity Rz in the z direction tends to be different.The resistivity in a direction parallel to the formation plane (i.e.,the x-y plane) is known as the horizontal resistivity, Rh, and theresistivity in the direction perpendicular to the plane of the formation(i.e., the z direction) is known as the vertical resistivity, Rv. Theindex of anisotropy, η, is defined as η=[Rv/Rh]^(1/2).

As a further complication to measuring formation resistivity, boreholesare generally perpendicular to formation beds. The angle between theaxis of the well bore and the orientation of the formation beds (asrepresented by the normal vector) has two components. These componentsare the dip angle and the strike angle. The dip angle is the anglebetween the borehole axis and the normal vector for the formation bed.The strike angle is the direction in which the borehole's axis “leansaway from” the normal vector.

Electromagnetic resistivity logging measurements are a complex functionof formation resistivity, formation anisotropy, and the formation dipand strike angles, which may all be unknown. Logging tools that fail toaccount for one or more of these parameters may produce inaccuratemeasurements. Moreover, tools that are able to provide dip and strikemeasurements along with azimuthal orientation information can be used toadjust the drilling direction to increase the borehole's exposure to ahydrocarbon bearing formation (“geosteering”). Specifically, it isdesirable to be able to steer a tool to an underground target using theinformation available from the logging tool. Moreover, it is desirableto be able to match raw measurements to the modeled response for thesystem to evaluate the system performance.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an illustrative logging while drilling environmentincluding dipping formation beds;

FIG. 2 shows an illustrative wireline logging environment includingdipping formation beds;

FIG. 3 shows a relationship between the orientation of a borehole and adipping formation bed;

FIG. 4 shows a hypothetical antenna arrangement for a tool having anorthogonal triaxial transmitter and two orthogonal triaxial receivers;

FIG. 5 shows angles for defining the orientation of a tilted antenna;

FIG. 6 shows a block diagram of an exemplary electronics module for anelectromagnetic resistivity tool;

FIG. 7 shows an illustrative electromagnetic resistivity logging toolhaving tilted transmitter and receiver antennas;

FIGS. 8a and 8b show an illustrative configuration of an antenna systemequipped with a tilted transmitter and a tilter receiver.

FIGS. 9a, 9b, and 9c show an illustrative configuration of a rotatingtool's relationship to surrounding environments.

FIG. 10 shows an illustrative Cartesian coordinate system that isrotated along z-directional axis with a clockwise angle φ_(t), in thex-y plane;

FIG. 11 shows an illustrative configuration of tool bin positions andcorresponding azimuthal angles;

FIG. 12 shows a flowchart of a processing scheme in accordance with anexemplary embodiment of the present invention;

FIG. 13 shows an illustrative configuration of an antenna system withtwo transmitters and one receiver;

FIG. 14 shows an illustrative configuration of a formation model of twolayered isotropic media.

FIG. 15 shows illustrative signal responses with azimuthal angle of 30°to the formation boundary.

FIG. 16 shows illustrative signal responses with x-axis of toolcoordinate system pointing to the formation boundary.

FIG. 17 shows illustrative signal responses with x-axis of toolcoordinate system parallel to the formation boundary.

FIG. 18 shows illustrative raw measurements of ADR-TT in a water tankwith a surrounding casing target.

FIG. 19 shows illustrative processed signals of ADR-TT in a water tankto steer the surrounding casing.

DETAILED DESCRIPTION

The terms “couple” or “couples,” as used herein are intended to meaneither an indirect or direct connection. Thus, if a first device couplesto a second device, that connection may be through a direct connection,or through an indirect electrical connection via other devices andconnections. The term “upstream” as used herein means along a flow pathtowards the source of the flow, and the term “downstream” as used hereinmeans along a flow path away from the source of the flow. The term“uphole” as used herein means along the drillstring or the hole from thedistal end towards the surface, and “downhole” as used herein meansalong the drillstring or the hole from the surface towards the distalend.

It will be understood that the term “oil well drilling equipment” or“oil well drilling system” is not intended to limit the use of theequipment and processes described with those terms to drilling an oilwell. The terms also encompass drilling natural gas wells or hydrocarbonwells in general. Further, such wells can be used for production,monitoring, or injection in relation to the recovery of hydrocarbons orother materials from the subsurface.

The present application discloses processing schemes for a rotatingelectromagnetic tool equipped with tilt antenna systems having arbitrarytilted angles for transmitters and receivers. Accordingly, the methodsdisclosed herein provide a novel approach to steering an undergroundtarget surrounding the electromagnetic tool. A relative azimuthal anglesensitivity of the tool is introduced and various mathematical relationsof tool signal responses are discussed upon a defined relative azimuthalangle between the tool and the surrounding target. By finding therelative azimuthal angle, one can steer the tool to its surroundingtarget as well as match raw measurements to the forwarding modelresponses.

Turning now to FIG. 1, an illustrative logging while drilling (“LWD”)environment is shown. A drilling platform 2 supports a derrick 4 havinga traveling block 6 for raising and lowering a drill string 8. A kelly10 supports the drill string 8 as it is lowered through a rotary table12. A drill bit 14 is driven by a downhole motor and/or rotation of thedrill string 8. As bit 14 rotates, it creates a borehole 16 that passesthrough various formations 18. A pump 20 may circulate drilling fluidthrough a feed pipe 22 to kelly 10, downhole through the interior ofdrill string 8, through orifices in drill bit 14, back to the surfacevia the annulus around drill string 8, and into a retention pit 24. Thedrilling fluid transports cuttings from the borehole into the pit 24 andaids in maintaining the borehole integrity.

An electromagnetic resistivity logging tool 26 may be integrated intothe bottom-hole assembly near the bit 14. As the bit extends theborehole through the formations, logging tool 26 collects measurementsrelating to various formation properties as well as the tool orientationand position and various other drilling conditions. The orientationmeasurements may be performed using an azimuthal orientation indicator,which may include magnetometers, inclinometers, and/or accelerometers,though other sensor types such as gyroscopes may be used in someembodiments, the tool includes a 3-axis fluxgate magnetometer and a3-axis accelerometer. The logging tool 26 may take the form of a drillcollar, i.e., a thick-walled tubular that provides weight and rigidityto aid the drilling process. A telemetry sub 28 may be included totransfer tool measurements to a surface receiver 30 and to receivecommands from the surface receiver 30.

In one embodiment, rotational position indicator array may contain botha 3-axis fluxgate magnetometer and a 3-axis accelerometer. As would beappreciated by those of ordinary skill in the art, with the benefit ofthis disclosure, the combination of those two sensor systems enables themeasurement of the tool face, inclination, and azimuth orientationangles of the borehole. The tool face and hole inclination angles arecalculated from the accelerometer sensor output. The magnetometer sensoroutputs are used to calculate the hole azimuth. With the tool face, thehole inclination, and the hole azimuth information, a tool in accordancewith the present invention can be used to steer the bit to the desirablebed. Specifically, the response difference or the response ratio can beused effectively to enter a desired payzone or to stay within thepayzone of interest.

At various times during the drilling process, the drill string 8 may beremoved from the borehole as shown in FIG. 2. Once the drill string hasbeen removed, logging operations can be conducted using a wirelinelogging tool 34, i.e., a sensing instrument sonde suspended by a cablehaving conductors for transporting power to the tool and telemetry fromthe tool to the surface. A resistivity imaging portion of the loggingtool 34 may have centralizing arms 36 that center the tool within theborehole as the tool is pulled uphole. A logging facility 44 may collectmeasurements from the logging tool 34, and may include computingfacilities for processing and storing the measurements gathered by thelogging tool.

Returning now to FIG. 1, it shows that the formations 18 are notperpendicular to the borehole, which may occur naturally or due todirectional drilling operations. The borehole may have a Cartesiancoordinate system 50 defined in accordance with the borehole's long axis(the z-axis) and the north side (or alternatively, the high side) of thehole (the x-axis). The formations 18, when characterized as a plane, mayhave a Cartesian coordinate system 51 defined in accordance with thenormal to the plane (the z″-axis) and the direction of steepest descent(the x″-axis). As shown in FIG. 3, the two Cartesian coordinate systemsare related by two rotations. Beginning with the borehole's Cartesiancoordinate system (x,y,z), a first rotation of angle γ is made about thez-axis. The resulting Cartesian coordinate system is denoted (x′,y′,z′).Angle γ is the relative strike angle, which indicates the direction ofthe formation dip relative to the borehole's Cartesian coordinatesystem. A second rotation of angle α is then made about the y′ axis.This aligns the borehole Cartesian coordinate system with the formationCartesian coordinate system. Angle α is the relative dip angle, which isthe slope angle of the beds relative to the long axis of the borehole.

The vertical resistivity is generally found to be the resistivity asmeasured perpendicular to the plane of the formation, and the horizontalresistivity is the resistivity as measured within the plane of theformation. Determination of each of these parameters (dip angle, strikeangle, vertical resistivity, and horizontal resistivity) is desirable.

FIG. 4 shows a hypothetical antenna configuration for a multi-componentelectromagnetic resistivity logging tool which may be embodied as awireline tool as well as a logging while drilling tool. A triad oftransmitter coils T_(X), T_(Y), and T_(Z), each oriented along arespective axis, may be provided. At least one triad of similarlyoriented receiver coils R_(1X), R_(1Y), and R_(1Z) may also be provided.For received signal measurements relative to the amplitude and phase ofthe transmitted signal (sometimes called “absolute” measurements) onlyone receiver triad would be used. A second triad of similarly orientedreceiver coils pairs R_(2X), R_(2Y), and R_(2Z) may also be providedwhen differential measurements are desired (e.g., a signal amplituderatio or a phase difference between receiver coils oriented along agiven axis). Differential measurements may offer increased spatialresolution.

Moran and Gianzero, in “Effects of Formation Anisotropy on ResistivityLogging Measurements” Geophysics, Vol. 44, No. 7, p. 1266 (1979), notedthat the magnetic field h in the receiver coils can be represented interms of the magnetic moments m at the transmitters and a couplingmatrix C:

h=Cm  (1)

In express form, equation (1) is:

$\begin{matrix}{\begin{bmatrix}{Hx} \\{Hy} \\{H\; z}\end{bmatrix} = {\begin{bmatrix}{Cxx} & {Cxy} & {Cxz} \\{Cyx} & {Cyy} & {Cyz} \\{Czx} & {Czy} & {Czz}\end{bmatrix}\begin{bmatrix}{M\; x} \\{My} \\{Mz}\end{bmatrix}}} & (2)\end{matrix}$

where Mx, My, and Mz are the magnetic moments (proportional to transmitsignal strength) created by transmitters Tx, Ty, and Tz, respectively.Hx, Hy, Hz are the magnetic fields (proportional to received signalstrength) at the receiver antennas Rx, Ry, and Rz, respectively.

In the antenna configuration of FIG. 4, if each transmitter is fired inturn, and signal measurements are made at each receiver in response toeach firing, nine absolute or differential measurements are obtained.These nine measurements enable the determination of a complete couplingmatrix C. (C_(IJ)=a_(IJ)v_(I) ^(J), where I is the index for receiverRx, Ry, or Rz, J is the index for transmitter Tx, Ty, or Tz, a_(IJ) is aconstant determined by the tool design, and v_(I) ^(J) is a complexvalue representing the signal amplitude and phase shift measured byreceiver I in response to the firing of transmitter J.). Knowledge ofthe complete coupling matrix enables the determination of dip angle,strike angle, vertical resistivity, and horizontal resistivity. A numberof techniques may be used to determine these parameters. For example,dip and strike angle may be determined from coupling matrix values asexplained by Li Gao and Stanley Gianzero, U.S. Pat. No. 6,727,706,“Virtual Steering of Induction Tool for Determination of Formation DipAngle.” Given these angles, vertical and horizontal resistivity can bedetermined in accordance with equations provided by Michael Bittar, U.S.Pat. No. 7,019,528 “Electromagnetic Wave Resistivity Tool Having aTilted Antenna for Geosteering Within a Desired Payzone.” Alternatively,a simultaneous solution for these parameters may be found as describedin the Bittar reference.

FIG. 5 shows two angles that may be used to specify the orientation of acoil antenna. The coil antenna may be considered as residing in a planehaving a normal vector. Tilt angle θ is the angle between thelongitudinal axis of the tool and the normal vector. Azimuth angle β isthe angle between the projection of the normal vector in the X-Y planeand the tool scribe line. Alternatively, in the downhole context,azimuthal angle β may represent the angle between projection of thenormal vector in the X-Y plane and the x-axis of the borehole Cartesiancoordinate system. As would be appreciated by those of ordinary skill inthe art, with the benefit of this disclosure, the methods and systemsdisclosed herein are not limited to any particular azimuthal angle.Specifically, the transmitter and receiver coils may have any azimuthalangle suitable for a particular application. Moreover, although thepresent application discloses an embodiment with a transmitter coilhaving the same azimuthal angle as a receiver coil, as would beappreciated by those of ordinary skill in the art, with the benefit ofthis disclosure, the methods and systems disclosed herein may also beapplied in instances where the transmitter coil(s) and the receivercoil(s) have differing azimuthal angles. For instance, in one exemplaryembodiment, one or both of the transmitter coil(s) and the receivercoil(s) may be positioned in a window inside the tool facing outwardsrather than being wrapped around the tool 902. Further, in oneembodiment, the normal vector of the coil antenna may be co-planar withthe X-Y plane.

It is noted that three transmitter antenna orientations and threereceiver antenna orientations are employed in the antenna configurationof FIG. 4. It has been discovered that when tool rotation is exploited,it is possible to determine the full coupling matrix with only onetransmitter and two receiver antenna orientations (or equivalently, onereceiver and two transmitter antenna orientations). Moreover, withcertain assumptions about the configuration of the formation, onetransmitter and receiver antenna orientation may be sufficient.

Before considering various tools having specific antenna configurations,the electronics common to each tool are described. FIG. 6 shows afunctional block diagram of the electronics for a resistivity tool. Theelectronics include a control module 602 that is coupled to an analogswitch 604. Analog switch 604 is configured to drive any one of thetransmitter coils T₁, T₂, T₃, T₄ with an alternating current (AC) signalfrom a signal source 606. In some embodiments, the signal sourceprovides radio frequency signals. The control module 602 preferablyselects a transmitter coil, pauses long enough for transients to dieout, then signals data storage/transmit module 610 to accept anamplitude and phase sample of the signals received by each of thereceivers. The control module 602 preferably repeats this processsequentially for each of the transmitters. The amplitude and phase shiftvalues are provided by amplitude and phase shift detector 608 which iscoupled to each of the receiver coils R₁ and R₂ for this purpose.

Control module 602 may process the amplitude and phase shiftmeasurements to obtain compensated measurements and/or measurementaverages. The raw, compensated, or averaged measurements, may betransmitted to the surface for processing to determine dip and strikeangles, vertical and horizontal resistivity, and other information suchas (i) distance to nearest bed boundary, (ii) direction of nearest bedboundary, and (iii) resistivity of any nearby adjacent beds. The datastorage/transmitter module 610 may be coupled to telemetry unit 28(FIG. 1) to transmit signal measurements to the surface. Telemetry unit28 can use any of several known techniques for transmitting informationto the surface, including but not limited to, (1) mud pressure pulse;(2) hard-wire connection; (3) acoustic waves; and (4) electromagneticwaves.

FIG. 7 shows an electromagnetic resistivity logging tool 702 having onlytwo receiver antenna orientations. The tool 702 is provided with one ormore regions 706 of reduced diameter. A wire coil 704 is placed in theregion 706 and in some embodiments is spaced away from the surface ofsubassembly 702 by a constant distance. To mechanically support andprotect the coil 704, a non-conductive filler material (not shown) suchas epoxy, rubber, or ceramic may be used in the reduced diameter regions706. Coil 704 is a transmitter coil, and coils 710 and 712 are receivingcoils. In operation, transmitter coil 704 transmits an interrogatingelectromagnetic signal which propagates through the borehole andsurrounding formation. Receiver coils 710, 712 detect the interrogatingelectromagnetic signal and provide a measure of the electromagneticsignal's amplitude attenuation and phase shift. For differentialmeasurements, additional receiver coils parallel to coils 710, 712 maybe provided at an axially spaced distance. From the absolute ordifferential amplitude attenuation and phase shift measurements, thecoupling matrix components can be determined and used as the basis fordetermining formation parameters and as the basis for geosteering.

In one embodiment, the transmitter coil 704 may be spaced approximately30 inches from the receiver coils 710, 712. The transmitter and receivercoils may comprise as little as one loop of wire, although more loopsmay provide additional signal power. The distance between the coils andthe tool surface is preferably in the range from 1/16 inch to ¾ inch,but may be larger. Transmitter coil 704 and receiver coil 712 may eachhave a tilt angle of about 45° and be aligned with the same azimuthalangle, while receiver coil 710 may have a tilt angle of about 45° and anazimuthal angle of 180° apart from receiver coil 712 (or equivalently, atilt angle of minus 45° at the same azimuth angle as receiver coil 712).

The signal measured by a tilted receiver in response to the firing of atilted transmitter can be expressed in terms of the signals v_(I) ^(J)that would be measured by the tool of FIG. 4. Specifically, FIGS. 8a and8b depict a configuration of an antenna system equipped with a tiltedtransmitter 802 and a tilted receiver 804 in accordance with anembodiment of the present invention. As shown in FIG. 8a , the Cartesiancoordinate system may be divided into 4 quadrants.

An electromagnetic resistivity logging tool may then be provided whichmay include a rotational position sensor. The electromagneticresistivity logging tool may further include a first transmitter antennaoriented in the first quadrant. A receiver antenna may be oriented inthe first quadrant or the third quadrant which is located diagonal tothe first quadrant. A second transmitter may be oriented in one of thesecond quadrant or the fourth quadrant. As shown in FIG. 8a , each ofthe second quadrant and the fourth quadrant are located adjacent to thefirst quadrant. With substantially same distance between the firsttransmitter antenna and the receiver antenna and the second transmitterantenna and the receiver antenna, the following steps may be performedas discussed in more detail below. First, an expression of rawmeasurements at the receiver antenna in response to the firing of thefirst transmitter may be obtained as a first expression. Next, anexpression of raw measurements at the receiver antenna in response tofiring of the second transmitter may be determined as a secondexpression. The first expression and the second expression may then beused to obtain an expression for the processed signals matching toforward model responses of the system.

When both transmitter and receiver coils are oriented at the sameazimuth angle β, the tilted receiver signal V^(T) _(R) is

$\begin{matrix}{{V_{R}^{T}(\beta)} = {{{\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta} \\{\sin \; \theta_{t}\sin \; \beta} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}\begin{bmatrix}v_{x}^{x} & v_{y}^{x} & v_{z}^{x} \\v_{x}^{y} & v_{y}^{y} & v_{z}^{y} \\v_{x}^{z} & v_{y}^{z} & z_{z}^{z}\end{bmatrix}}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta} \\{\sin \; \theta_{r}\sin \; \beta} \\{\cos \; \theta_{r}}\end{bmatrix}} = {\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta} \\{\sin \; \theta_{t}\sin \; \beta} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}{v_{M}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta} \\{\sin \; \theta_{r}\sin \; \beta} \\{\cos \; \theta_{r}}\end{bmatrix}}}}} & (3)\end{matrix}$

where, θ_(t) is the tilt angle of the transmitter related to the z-axiswhich is denoted by line 800 of FIG. 8 b; θ _(r) is the tilt angle ofthe receiver related to the z-axis which is denoted by line 800; v_(I)^(J) is a complex value representing the signal amplitude and phaseshift measured by the receiver 804 in the I-directional dipole inresponse to firing of transmitter 802 in the J-directional dipole; andv_(M) is the 3×3 complex voltage matrix corresponding to v_(I) ^(J). Aswould be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, v_(I) ^(J) in Eq. (3) is affected byenvironmental conditions. Specifically, v_(I) ^(J) in Eq. (3) may beaffected by two environmental conditions, a surrounding target such as acasing in a homogeneous isotropic medium and a surrounding boundary.

FIG. 9a depicts configuration of a rotating tool's relationship tosurrounding casing and FIG. 9b depicts configuration of a rotatingtool's relationship to a surrounding boundary and FIG. 9c depictsconfiguration of a rotating tool's relationship to electrical anisotropyof a thinly laminated formation.

With reference to FIG. 9c , electrical anisotropy exists in laminatedthin layers 910 each with different resistivity values, producing ahigher resistivity in the direction perpendicular to the fracture plan(vertical resistivity Rv, as shown on axis 912) than the resistivity(horizontal resistivity Rh, as shown on axis 912) in the paralleldirection. While operating the tool 902 downhole in an anisotropicformation, the tool's 902 highside may not point to the formation planewhere horizontal resistivity Rh exists. Therefore, measurements will beaffected by the azimuth difference φ_(t) between the tool's 902 highside and the direction pointing to the formation plane 910.

As shown in FIGS. 9a, 9b and 9c , all environments present a relativeazimuthal angle φ_(t) between the coordinate defined by the tool 902 andsurrounding environments. While operating a tool 902 downhole, az-directional axis of the tool's Cartesian coordinate system 904 may beselected based on the tool's current borehole path. Similarly, anazimuthal angle β for the tool may be determined in the x-y plane of thetool Cartesian coordinate system 904. Specifically, the x-directionalaxis may be defined in the high side of tool 902 based on magnetometerand/or gravity system of the tool 902 with corresponding azimuthal angleof zero. Accordingly, as depicted in FIGS. 9a, 9b, and 9c , the angleφ_(t) between high side direction of the tool 902 and the direction withthe closest distance (L) to the casing position 906 as shown in FIG. 9a, a boundary plane 908 as shown in FIG. 9b , or a thinly laminatedformation 910 is herein defined as relative azimuthal angle φ_(t).

In order to determine the relative azimuthal angle φ_(t), the high sideof the tool 902 may be hypothetically rotated along with the z-axis ofthe tool Cartesian coordinate system 904 toward the target in FIG. 9.Accordingly, by applying the rotating angle φ_(t) into Eq. (3), themeasured angle in response to the new azimuthal angle β′ defined in therotated Cartesian coordinate system denoted x′, y′, and z′ as shown inFIG. 10 may be given by

$\begin{matrix}{{V_{R}^{T}\left( \beta^{\prime} \right)} = {{{\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta^{\prime}} \\{\sin \; \theta_{t}\sin \; \beta^{\prime}} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}\begin{bmatrix}v_{x}^{\prime \; x} & v_{y}^{\prime \; x} & v_{z}^{\prime \; x} \\v_{x}^{\prime \; y} & v_{y}^{\prime \; y} & v_{z}^{\prime \; y} \\v_{x}^{\prime \; z} & v_{y}^{\prime \; z} & v_{z}^{\prime \; z}\end{bmatrix}}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta^{\prime}} \\{\sin \; \theta_{r}\sin \; \beta^{\prime}} \\{\cos \; \theta_{r}}\end{bmatrix}} = {\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta^{\prime}} \\{\sin \; \theta_{t}\sin \; v} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}{v_{M}^{\prime}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta^{\prime}} \\{\sin \; \theta_{r}\sin \; \beta^{\prime}} \\{\cos \; \theta_{r}}\end{bmatrix}}}}} & (4)\end{matrix}$

where β′ equals (β+φ_(t)) and v′_(M) is a 3×3 complex voltage matrixcorresponding to v′_(I) ^(J) measured in the rotated Cartesiancoordinate system or a new Cartesian coordinate system with thex-directional axis pointing to the surrounding target. Specifically, therelationship when the Cartesian coordinate system is rotated along withthe z-directional axis with a clockwise relative azimuthal angle φ_(t)in the x-y plane may be characterized as shown in FIG. 10.

Because the high side of the tool 902 points to the target after therotation of the Cartesian coordinate system 904 as shown in FIG. 10,based on the electromagnetic concept, Equation (4) may be simplified as:

$\begin{matrix}{{V_{R}^{T}\left( \beta^{\prime} \right)} = {{{\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta^{\prime}} \\{\sin \; \theta_{t}\sin \; \beta^{\prime}} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}\begin{bmatrix}v_{x}^{\prime \; x} & 0 & v_{z}^{\prime \; x} \\0 & v_{y}^{y} & 0 \\v_{x}^{\prime \; z} & 0 & v_{z}^{\prime \; z}\end{bmatrix}}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta^{\prime}} \\{\sin \; \theta_{r}\sin \; \beta^{\prime}} \\{\cos \; \theta_{r}}\end{bmatrix}} = {\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta^{\prime}} \\{\sin \; \theta_{t}\sin \; \beta^{\prime}} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}{v_{M}^{\prime}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta^{\prime}} \\{\sin \; \theta_{r}\sin \; \beta^{\prime}} \\{\cos \; \theta_{r}}\end{bmatrix}}}}} & (5)\end{matrix}$

The relationship between Eq. (3) and Eq. (5) is the rotation of theCartesian coordinate system 904 of the tool 902 along the z-directionalaxis and may be described by the relative azimuthal angle φ_(t) as shownin FIG. 10. Based on the relationship shown in FIG. 10, v′_(M) may beobtained as:

$\begin{matrix}{v_{M}^{\prime} = {\begin{bmatrix}{\cos \; \varphi_{t}} & {\sin \; \varphi_{t}} & 0 \\{{- \sin}\; \varphi_{t}} & {\cos \; \varphi_{t}} & 0 \\0 & 0 & 1\end{bmatrix}{v_{M}\begin{bmatrix}{\cos \; \varphi_{t}} & {\sin \; \varphi_{t}} & 0 \\{{- \sin}\; \varphi_{t}} & {\cos \; \varphi_{t}} & 0 \\0 & 0 & 1\end{bmatrix}}}} & (6)\end{matrix}$

As would be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, if only one surrounding target isconsidered, four equations related to the measured complex voltagecomponents v_(I) ^(J) may be derived as shown below:

v _(x) ^(y) +v _(y) ^(x)=0  (7a)

(v _(x) ^(x) +v _(y) ^(y))cos φ_(t) sin φ_(t) +v _(x) ^(y) sin^(2 φ)_(t) +v _(y) ^(x) cos^(2 φ) _(t)=0  (7b)

v _(x) ^(z) sin φ_(t) +v _(y) ^(z) cos φ_(t)=0  (7c)

−v _(z) ^(x) sin φ_(t) +v _(z) ^(y) cos φ_(t)=0  (7d)

In order to analyze Eq. (7), two conditions may be taken intoconsideration. The first assumed condition is instances where therotation angle φ_(t) is assumed to be either π/2 (90°) or 3π/2 (270°).Under the first assumed condition from Eq. (7), it may be concluded thatv_(y) ^(x)=v_(x) ^(y)=v_(z) ^(x)=v_(x) ^(z)=0. Therefore, the measuredraw signal presented in Eq. (3) may be expressed as:

$\begin{matrix}\begin{matrix}{{V_{R}^{T}(\beta)} = {{\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta} \\{\sin \; \theta_{t}\sin \; \beta} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}\begin{bmatrix}v_{x}^{x} & 0 & 0 \\0 & v_{y}^{y} & v_{z}^{y} \\0 & v_{y}^{z} & v_{z}^{z}\end{bmatrix}}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta} \\{\sin \; \theta_{r}\sin \; \beta} \\{\cos \; \theta_{r}}\end{bmatrix}}} \\{= {{\left( {{\frac{v_{x}^{x}}{2}\sin \; \theta_{t}\sin \; \theta_{r}} - {\frac{v_{y}^{y}}{2}\sin \; \theta_{t}\sin \; \theta_{r}}} \right)\cos \; 2\beta} +}} \\{{{\left( {{v_{y}^{z}\cos \; \theta_{t}\sin \; \theta_{r}} + {v_{z}^{y}\sin \; \theta_{t}\cos \; \theta_{r}}} \right)\sin \; \beta} +}} \\{\left( {{\frac{v_{x}^{x}}{2}\sin \; \theta_{t}\sin \; \theta_{r}} + {\frac{v_{y}^{y}}{2}\sin \; \theta_{t}\sin \; \theta_{r}} + {v_{z}^{z}\cos \; \theta_{t}\cos \; \theta_{r}}} \right)} \\{= {{V_{1}\cos \; 2\beta} + {V_{2}\sin \; \beta} + V_{3}}}\end{matrix} & (8)\end{matrix}$

where V₁, V₂, and V₃ may be determined by current environments and tool902 antenna structures as:

$V_{1} = {{\frac{v_{x}^{x}}{2}\sin \; \theta_{t}\sin \; \theta_{r}} - {\frac{v_{y}^{y}}{2}\sin \; \theta_{t}\sin \; \theta_{r}}}$V₂ = v_(y)^(z)cos  θ_(t)sin  θ_(r) + v_(z)^(y)sin  θ_(t)cos  θ_(r)$V_{3} = {{\frac{v_{x}^{x}}{2}\sin \; \theta_{t}\sin \; \theta_{r}} + {\frac{v_{y}^{y}}{2}\sin \; \theta_{t}\sin \; \theta_{r}} + {v_{z}^{z}\cos \; \theta_{t}\cos \; \theta_{r}}}$

On the other hand, under a second assumed condition, if the rotationangle φ_(t) is neither π/2 (90°) nor 3π/2 (270°), the followingrelationships may be derived from Eq. (7):

$\begin{matrix}{v_{x}^{y} = {- v_{y}^{x}}} & \left( {9a} \right) \\\left\{ \begin{matrix}{{v_{y}^{x} = {{- \frac{1}{2}}{\tan \left( {2\varphi_{t}} \right)}\left( {v_{x}^{x} + v_{y}^{y}} \right)}},} & {{{if}\mspace{14mu} {\cos \left( {2\varphi_{t}} \right)}} \neq 0} \\{{v_{x}^{x} = {- v_{y}^{y}}},} & {{{if}\mspace{14mu} {\cos \left( {2\varphi_{t}} \right)}} = 0}\end{matrix} \right. & \left( {9b} \right) \\{v_{y}^{z} = {{- v_{x}^{z}}\tan \; \varphi_{t}}} & \left( {9c} \right) \\{v_{z}^{y} = {v_{z}^{x}\tan \; \varphi_{t}}} & \left( {9d} \right)\end{matrix}$

and therefore, the measured raw signal may be modified as:

                                                              (10)$\begin{matrix}{{V_{R}^{T}(\beta)} = \left\{ \begin{matrix}{{{\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta} \\{\sin \; \theta_{t}\sin \; \beta} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}\begin{bmatrix}v_{x}^{x} & {{- \frac{1}{2}}{\tan \left( {2\varphi_{t}} \right)}\left( {v_{x}^{x} + v_{y}^{y}} \right)} & v_{z}^{x} \\{\frac{1}{2}{\tan \left( {2\varphi_{t}} \right)}\left( {v_{x}^{x} + v_{y}^{y}} \right)} & v_{y}^{y} & {v_{z}^{x}\tan \; \varphi_{t}} \\v_{x}^{z} & {{- v_{x}^{z}}\tan \; \varphi_{t}} & v_{z}^{z}\end{bmatrix}}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta} \\{\sin \; \theta_{r}\sin \; \beta} \\{\cos \; \theta_{r}}\end{bmatrix}},} & {{{if}\mspace{14mu} {\cos \left( {2\varphi_{t}} \right)}} \neq 0} \\{{{\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta} \\{\sin \; \theta_{t}\sin \; \beta} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}\begin{bmatrix}v_{x}^{x} & v_{y}^{x} & v_{z}^{x} \\{- v_{y}^{x}} & {- v_{y}^{x}} & {v_{z}^{x}\tan \; \varphi_{t}} \\v_{x}^{z} & {{- v_{x}^{z}}\tan \; \varphi_{t}} & v_{z}^{z}\end{bmatrix}}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta} \\{\sin \; \theta_{r}\sin \; \beta} \\{\cos \; \theta_{r}}\end{bmatrix}},} & {{{if}\mspace{14mu} {\cos \left( {2\varphi_{t}} \right)}} = 0}\end{matrix} \right.} \\{= {{\left( {\frac{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}}{2} - \frac{v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}{2}} \right)\cos \; 2\beta} + {\left( {{v_{x}^{z}\cos \; \theta_{t}\sin \; \theta_{r}} + {v_{z}^{x}\sin \; \theta_{t}\cos \; \theta_{r}}} \right)\cos \; \beta} +}} \\{{{\left( {{v_{z}^{x}\sin \; \theta_{t}\cos \; \theta_{r}} - {v_{x}^{z}\cos \; \theta_{t}\sin \; \theta_{r}}} \right)\tan \; \varphi_{t}\sin \; \beta} + \left( {\frac{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}}{2} + \frac{v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}{2} + {v_{z}^{z}\cos \; \theta_{t}\cos \; \theta_{r}}} \right)}} \\{= {{V_{1}\cos \; 2\beta} + {V_{4}\cos \; \beta} + {V_{5}\sin \; \beta} + V_{3}}}\end{matrix}$

Again, V₄ and V₅ may be determined by the existing environment and tool902 antennal structures where V₄=v_(x) ^(z) cos θ_(t) sin θ_(r)+v_(z)^(x) sin θ_(t) cos θ_(r) and V₅=(v_(z) ^(x) sin θ_(t) cos θ_(r)−z_(x)^(z) cos θ_(t) sin θ_(r))tan θ_(t).

Equations (8) and (10) provide a better understanding of rawmeasurements from a rotating tool equipped with a tilt antenna systemwith only one surrounding target. In order to simplify the analysis, theforwarding model normally only considers one target surrounding the tool902 with its high side pointing to that target. As a result, ininstances where there is more than one target surrounding the tool 902,and/or there is a significant relative azimuthal angle φ_(t), themodeling responses could explain real behaviors of tool measurements butnot get accurate inversion results. Consequently before invertingformation parameters based on raw measurements, it is desirable toprocess the raw measurements first to obtain better signal quality thatis closer to the modeling responses. Eq. (5) represents the modelingresponses and may be expressed as:

$\begin{matrix}{{V_{R}^{T}\left( \beta^{\prime} \right)} = {\left\lbrack {\left( {\frac{v_{x}^{\prime \; x}\sin \; \theta_{t}\sin \; \theta_{r}}{2} - \frac{v_{y}^{\prime \; y}\sin \; \theta_{t}\sin \; \theta_{r}}{2}} \right)\cos \; 2\; \beta^{\prime}} \right\rbrack + {\quad{\left\lbrack {\left( {{v_{x}^{\prime \; z}\cos \; \theta_{t}\sin \; \theta_{r}} + {v_{z}^{\prime \; x}\sin \; \theta_{t}\cos \; \theta_{r}}} \right)\cos \; \beta^{\prime}} \right\rbrack + \left( {{v_{z}^{\prime \; z}\cos \; \theta_{t}\cos \; \theta_{r}} + \frac{v_{x}^{\prime \; x}\sin \; \theta_{t}\sin \; \theta_{r}}{2} + \frac{v_{y}^{\prime \; y}\sin \; \theta_{t}\sin \; \theta_{r}}{2}} \right)}}}} & (11)\end{matrix}$

Based on Eq. (6) and Eqs. (7a)-(7d), if φ_(t) is π/2 (90°) or 3π/2(270°), Eq. (11) may be rewritten as:

$\begin{matrix}\begin{matrix}{{V_{R}^{T}\left( \beta^{\prime} \right)} = {\left\lbrack {\left( {\frac{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}}{2} - \frac{v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}{2}} \right)\cos \; 2\beta^{\prime}} \right\rbrack +}} \\{{\left\lbrack {\left( {{v_{z}^{y}\sin \; \theta_{t}\cos \; \theta_{r}} - {v_{y}^{z}\cos \; \theta_{t}\sin \; \theta_{r}}} \right)\sin \; \varphi_{t}\cos \; \beta^{\prime}} \right\rbrack +}} \\{\left( {{v_{z}^{z}\cos \; \theta_{t}\cos \; \theta_{r}} + \frac{{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}} + {v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}}{2\; \cos \; 2\varphi_{t}}} \right)} \\{= {{V_{1}\cos \; 2\beta^{\prime}} + {V_{6}\sin \; \varphi_{t}\cos \; \beta^{\prime}} + V_{7}}}\end{matrix} & (12)\end{matrix}$

where V₆=v_(z) ^(y) sin θ_(t) cos θ_(r)−v_(y) ^(x) cos θ_(t) sin θ_(r)and

$V_{7} = {{v_{z}^{z}\cos \; \theta_{t}\cos \; \theta_{r}} + {\frac{{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}} + {v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}}{2\; \cos \; 2\varphi_{t}}.}}$

In contrast, if the rotation angle φ_(t) is neither π/2 (90°) nor 3π/2(270°), Eq. (11) may be modified as:

$\begin{matrix}\begin{matrix}{{V_{R}^{T}\left( \beta^{\prime} \right)} = {\left\lbrack {\left( {\frac{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}}{2} - \frac{v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}{2}} \right)\cos \; 2\beta^{\prime}} \right\rbrack +}} \\{{\left\lbrack {\frac{\left( {{v_{x}^{z}\cos \; \theta_{t}\sin \; \theta_{r}} + {v_{z}^{x}\sin \; \theta_{t}\cos \; \theta_{r}}} \right)}{\cos \; \varphi_{t}}\cos \; \beta^{\prime}} \right\rbrack +}} \\{\left( {{v_{z}^{z}\cos \; \theta_{t}\cos \; \theta_{r}} + \frac{{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}} + {v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}}{2\; \cos \; 2\varphi_{t}}} \right)} \\{{= {{V_{1}\cos \; 2\beta^{\prime}} + {\frac{V_{4}}{\cos \; \varphi_{t}}\cos \; \beta^{\prime}} + V_{7}}},{{{if}\mspace{14mu} {\cos \left( {2\varphi_{t}} \right)}} \neq 0}}\end{matrix} & \left( {13a} \right) \\\begin{matrix}{{V_{R}^{T}\left( \beta^{\prime} \right)} = {\left\lbrack {\left( {\frac{v_{x}^{x}\sin \; \theta_{t}\sin \; \theta_{r}}{2} - \frac{v_{y}^{y}\sin \; \theta_{t}\sin \; \theta_{r}}{2}} \right)\cos \; 2\beta^{\prime}} \right\rbrack +}} \\{{\left\lbrack {\frac{\left( {{v_{x}^{z}\cos \; \theta_{t}\sin \; \theta_{r}} + {v_{z}^{x}\sin \; \theta_{t}\cos \; \theta_{r}}} \right)}{\cos \; \varphi_{t}}\cos \; 2\beta^{\prime}} \right\rbrack +}} \\{\left( {{v_{z}^{z}\cos \; \theta_{t}\cos \; \theta_{r}} + {v_{x}^{y}\sin \; \theta_{t}\sin \; \theta_{r}\sin \; 2\varphi_{t}}} \right)} \\{{= {{V_{1}\cos \; 2\beta^{\prime}} + {\frac{V_{4}}{\cos \; \varphi_{t}}\cos \; \beta^{\prime}} + V_{8}}},{{{if}\mspace{14mu} {\cos \left( {2\varphi_{t}} \right)}} = 0}}\end{matrix} & \left( {13b} \right)\end{matrix}$

where V₈=v_(z) ^(z) cos θ_(t) cos θ_(r)+v_(x) ^(y) sin θ_(t) sin θ_(r)sin 2φ_(t).

Accordingly, Eq. (8) and Eq. (10) present signal behaviors of raw toolmeasurements in an environment condition of a relative azimuthal angleφ_(t) toward surrounding target, whereas Eq. (12) and Eqs. (13a) and(13b) demonstrate how forwarding model signals are presented by themeasured raw signals. Consequently, these equations provide processingschemes on raw measurements to match to forwarding model responses. Aswould be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, the relationships identified in Equations(8), (10), (12) and (13a)-(13b) may be used to reveal that the amplitudeof double sine wave responses is consistent. Accordingly, theseequations may be used to match raw measurements to forwarding modelresponses.

FIG. 11 depicts the configuration of the bin positions and correspondingazimuthal angles of a tool 902 in accordance with an embodiment of thepresent invention. As shown in FIG. 11, N is the total number of bins inthe rotating tool 902 and a bin i with azimuthal angle β_(i) and a bin jwith azimuthal angle β_(j) are located opposite each other so thatβ_(i)=β_(j)+180°.

FIG. 12 depicts a flow chart for the proposed processing scheme of theraw measurements in accordance with an embodiment of the presentinvention. First, at step 1202, raw measurements are obtained from thetool 902. As would be appreciated by those of ordinary skill in the art,with the benefit of this disclosure, obtaining raw measurements from thetool 902 is well known in the art and will therefore not be discussed indetail herein. Once the raw measurements are obtained, they may beprocessed in accordance with an embodiment of the present invention.

The calculations in accordance with an embodiment of the presentinvention may be simplified by assuming that the rotating angle φ_(t) isneither π/2 (90°) nor 3π/2 (270°) and accordingly, cos(2φ_(t))≠0. Aswould be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, this assumption does not exclude otherrotating angles from the proposed processing scheme and is merely usedto simplify the conditions to obtain the following general expressionsfor the proposed processing scheme. Accordingly, the methods disclosedherein are applicable to any rotating angle in implementation. Under theassumed condition that the rotating angle φ_(t) is neither π/2 (90°) nor3π/2 (270°) and using a tool as shown in FIG. 11, three steps may beused to analyze the raw measurements in step 1204. In the first step,all the bin measurements are averaged as follows to obtain V_(step1):

$V_{{Step}\; 1} \equiv {\frac{1}{N}{\sum\limits_{i = 1}^{N}\; {V_{R}^{T}\left( \beta_{i} \right)}}}$

Next, step 2 entails averaging two raw complex voltage measurementswhere one is in a bin direction and the other is in the opposite bindirection. Specifically, V_(step2) may be determined as:

${{V_{{Step}\; 2}\left( \beta_{i} \right)} \equiv {\frac{{V_{R}^{T}\left( \beta_{i} \right)} + {V_{R}^{T}\left( \beta_{j} \right)}}{2} - V_{{step}\; 1}}},{i = 1},2,\ldots \mspace{14mu},N$

Finally, step 3 is averaging the subtraction of one raw measurement in abin direction from the other raw measurement in opposite bin directionto obtain V_(step3) as follows:

${V_{{Step}\; 3}\left( \beta_{i} \right)} \equiv \frac{{V_{R}^{T}\left( \beta_{i} \right)} - {V_{R}^{T}\left( \beta_{j} \right)}}{2}$

The three steps above may be used to distribute the raw measurements ofEq. (10) into three parts: (1) a complex voltage; (2) azimuthal voltagesas a sinusoid wave response with double periods; and (3) azimuthalvoltages as a sinusoid wave response with a single period with respectto tool's azimuthal angle (0˜360°), or the tool's bin 1 to bin N asshown in FIG. 11).

As would be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, both cosine wave fitting and sine wavefitting functions may be used to fit processed responses. Specifically,because cosine wave and sine wave responses are theoretically similarexcept for a 180° phase shift, the only difference between cosine andsine fitting functions is 180° phase shift on estimated φ_(t).Accordingly, the methods and systems disclosed herein may be carried outusing both sine wave or cosine wave fitting methods.

After applying the three steps, at step 1206, the second step with twoperiod sinusoid wave responses is examined. As discussed above, theamplitude of the cosine wave responses of the second step remains thesame, regardless of the relative azimuthal rotation angle φ_(t). As aresult, investigation of this part will provide a good estimated resultfor the relative azimuthal rotation angle φ_(t). Since the response atthis second step is a sinusoid wave with two period, there are fourrotation angles that may be found by cosine curve fitting. Cosine curvefitting is well known to those of ordinary skill in the art and willtherefore not be discussed in detail herein. After the cosine curvefitting is performed and four rotation angles are determined, at step1208, the process may be simplified by only considering the two smallestvalues obtained for the relative azimuthal rotation angle φ_(t), whereone of the two values could be the real value of φ_(t) and the otherwill be either (φ_(t)+90°) or (φ_(t)−90°) depending on which absolutevalue is smaller. Accordingly, at step 1208, the amplitude (V1 from Eq.(10)) corresponding to each of the values of the relative azimuthalrotation angle φ_(t) may be determined. Once the value of V1 (from Eq.(10)) and the two possible values of relative azimuthal rotation angleφ_(t) are known, at step 1210, a single period sinusoid wave isidentified. Next, cosine curve fitting is utilized and the two computedrelative azimuthal rotation angles φ_(t) may be used to calculate thecorresponding amplitude of V4 and V5 (from Eq. (10)). Specifically, thetwo possible values of the relative azimuthal rotation angle φ_(t) areapplied to step 1210 to identify the correct value of the relativeazimuthal rotation angle φ_(t) as well as V4 and V5 in Eq. (10). The twopossible values of the relative azimuthal rotation angle φ_(t) willcause different estimated values of V4 and V5 after cosine curve fittingbetween steps 1210 and 1212.

Next, at step 1212, an accurate value for the relative azimuthalrotation angle φ_(t), V1, V4, and V5 of Eq. (10) may be determined.Based on the distinct tilt angle designs of the antenna system, V4 maybe larger than V5 in amplitude for some tilt transmitter and tiltreceiver designs and V4 may be smaller than V5 in amplitude for othertilt antenna designs. Knowing the antenna structure in advance, one candiffer V4 and V5 in their amplitudes. Accordingly, an accurate relativeazimuthal rotation angle φ_(t) may be obtained by applying the twoestimated values found in the second step and the third step and thencomparing the corresponding amplitudes of V4 and V5. Specifically, thecorrect value of the relative azimuthal rotation angle φ_(t) may bedetermined when (1) the tilt angles of Tx and Rx are known; (2) twopossible values of the relative azimuthal rotation angle φ_(t) areidentified in step 1208; (3) V4 and V5 values are calculated based ontwo possible values of the relative azimuthal rotation angle φ_(t) and(4) the absolute amplitude of V4 and V5 are known and can be compared.

In one exemplary embodiment, a constant voltage of the first step isobtained at step 1214, which is the value of V3 in Eq. (10) at step1216. Next, at step 1216, the constant voltage of the third step may beused to determine the value of V3 in Eq. (10). Using the value of V3obtained in step 1216 in conjunction with the values obtained for V1,V4, V5 and φ_(t), the raw measurement may be curve-fitted and describedas Eq. 10 at step 1218.

In another exemplary embodiment, once the values for V1, V4, V5 andφ_(t) are determined, at step 1220 the signal's match with theforwarding model may be determined as shown in FIG. 12.

However, as would be appreciated by those of ordinary skill in the art,with the benefit of this disclosure, using a single tilttransmitter-receiver pair does not permit matching raw measurements ofEq. (10) to the modeling responses of Eq. (13a) because calculating V7in Eq. (13a) requires the evaluation of values of the measured voltagecomponents of V_(z) ^(z) and V_(x) ^(x)+V_(y) ^(y). In order to overcomethis problem, a two antenna system design with the same spacing betweentransmitter and receiver but different tilt angles of transmitter ordifferent tilt angles of receiver such as the one depicted in FIG. 13may be used. In one exemplary embodiment, a commercially availableazimuthal directional resistivity tool with a tilt transmitter and atilt receiver (ADR-TT) such as ones available from Halliburton EnergyServices of Duncan, Oklahoma, may be used.

As shown in FIG. 13, transmitters and receivers may be placed along anaxis 1300. A distance d between upper transmitter T_(up) and centralreceiver Rx may be the same as the distance between lower transmitterT_(dn) and central receiver Rx. However, the upper transmitter T_(up)and the central receiver RX are parallel to each other while the lowertransmitter T_(dn) and the central receiver Rx are perpendicular to eachother. Accordingly, the raw measured signals received in the centralreceiver Rx in response to firing of the upper transmitter T_(up) may berepresented as:

$\begin{matrix}{{V_{R_{x}}^{T_{up}}(\beta)} = {\left\lbrack {\left( {\frac{v_{x}^{x}}{4} - \frac{v_{y}^{y}}{4}} \right)\cos \; 2\left( {\beta^{\prime} - \varphi_{t}} \right)} \right\rbrack + \left\lbrack {{\frac{1}{2}\left( {v_{x}^{z} + v_{z}^{x}} \right){\cos \left( {\beta^{\prime} - \varphi_{t}} \right)}} + {\frac{1}{2}\left( {v_{z}^{x} - v_{x}^{z}} \right)\tan \; \varphi_{t}{\sin \left( {\beta^{\prime} - \varphi_{t}} \right)}}} \right\rbrack + \left( {\frac{v_{x}^{x}}{4} + \frac{v_{y}^{y}}{4} + \frac{v_{z}^{z}}{2}} \right)}} & \left( {14a} \right)\end{matrix}$

In contrast, the measured signals receiver in the center receiver Rx inresponse to firing of lower transmitter T_(dn) may be represented as:

$\begin{matrix}{{V_{R_{x}}^{T_{dn}}(\beta)} = {\left\lbrack {\left( {\frac{v_{x}^{x}}{4} - \frac{v_{y}^{y}}{4}} \right)\cos \; 2\left( {\beta^{\prime} - \varphi_{t}} \right)} \right\rbrack + \left\lbrack {{\frac{1}{2}\left( {v_{x}^{z} - v_{z}^{x}} \right){\cos \left( {\beta^{\prime} - \varphi_{t}} \right)}} - {\frac{1}{2}\left( {v_{z}^{x} + v_{x}^{z}} \right)\tan \; \varphi_{t}{\sin \left( {\beta^{\prime} - \varphi_{t}} \right)}}} \right\rbrack + \left( {\frac{v_{x}^{x}}{4} + \frac{v_{y}^{y}}{4} - \frac{v_{z}^{z}}{2}} \right)}} & \left( {14b} \right)\end{matrix}$

Accordingly, after performing the first step procedure on both Eq. (14a)and Eq. (14b), the constant complex voltages may be expressed as:

$\begin{matrix}{V_{{Step}\; 1{\_ TupRx}} = {\frac{v_{x}^{x}}{4} + \frac{v_{y}^{y}}{4} + \frac{v_{z}^{z}}{2}}} & \left( {15a} \right) \\{V_{{Step}\; 1{\_ TdnRx}} = {\frac{v_{x}^{x}}{4} + \frac{v_{y}^{y}}{4} - \frac{v_{z}^{z}}{2}}} & \left( {15b} \right)\end{matrix}$

Consequently, the processed signals matching to the forwarding modelresponses may be represented as:

$\begin{matrix}{{V_{R_{x}}^{T_{up}}\left( \beta^{\prime} \right)} = {\left\lbrack {\left( {\frac{v_{x}^{x}}{4} - \frac{v_{y}^{y}}{4}} \right)\cos \; 2\beta^{\prime}} \right\rbrack + \left\lbrack {\frac{1}{2\; \cos \; \varphi_{t}}\left( {v_{x}^{z} + v_{z}^{x}} \right)\cos \; \beta^{\prime}} \right\rbrack + \left( \frac{V_{{Step}\; 1{\_ TupRx}} - V_{{Step}\; 1{\_ TdnRx}}}{2} \right) + \left( \frac{V_{{Step}\; 1{\_ TupRx}} + V_{{Step}\; 1{\_ TdnRx}}}{2\; {\cos \left( {2\varphi_{t}} \right)}} \right)}} & \left( {16a} \right) \\{{V_{R_{x}}^{T_{dn}}\left( \beta^{\prime} \right)} = {\left\lbrack {\left( {\frac{v_{x}^{x}}{4} - \frac{v_{y}^{y}}{4}} \right)\cos \; 2\beta^{\prime}} \right\rbrack + \left\lbrack {\frac{1}{2\; \cos \; \varphi_{t}}\left( {v_{x}^{z} - v_{z}^{x}} \right)\cos \; \beta^{\prime}} \right\rbrack + \left( \frac{V_{{Step}\; 1{\_ TdnRx}} - V_{{Step}\; 1{\_ TupRx}}}{2} \right) + \left( \frac{V_{{Step}\; 1{\_ TdnRx}} + V_{{Step}\; 1{\_ TupRx}}}{2\; {\cos \left( {2\varphi_{t}} \right)}} \right)}} & \left( {16b} \right)\end{matrix}$

As would be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, the methods and systems disclosed herein areapplicable to antenna systems with arbitrary tilt angle for transmitters(Tx) and receivers (Rx). The systems and methods disclosed herein arenot limited to any specific antenna configuration and may be applied toa number of systems, including, but not limited to, antenna systemshaving one tilt Tx and one tilt Rx, combinations of two tilt Txs and oneRx, combinations of two tilt Rxs and one Tx, or combinations of multipletilt Txs and multiple tilt Rxs. Moreover, the tilt angles of thetransmitters and receivers may be the same or different. Further, aswould be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, based on the reciprocity theorem, antennasmay operate equally well as transmitters or receivers. Accordingly, anantenna may be applied as a transmitter in one implementation and as areceiver in another. As a result, all the configurations oftransmitters-receivers of the antenna systems disclosed herein may beinterchangeable. Specifically, transmitters may be used as receivers andreceivers may be used as transmitters.

EXAMPLE I

FIG. 14 depicts the configuration of a formation model of two layeredisotropic media with a boundary between the two layers 1402 and 1404. Inthis example, the upper layer 1402 has a resistivity of 1 Ω.m and thelower layer 1204 has a resistivity of 4 Ω.m. In the example, therelative dip angle is 15° and the reference point of ADR-TT tool isabout 3.58 ft. away from the boundary as shown in FIG. 14. FIG. 15depicts the modeling signals of Eq. (14a) and Eq. (14b) for thisformation model with a relative azimuthal rotation angle φ_(t) of 30°.The total 32 bin signals were calculated with a spacing of 52 inch(denoted as d in FIG. 13) and operating frequency of 125 kHz.

FIG. 16 signal response with x-axis of the tool Cartesian coordinatesystem 904 pointing to the formation boundary. Specifically, FIG. 16represents the signals of Eq. (14a) and Eq. (14b) if rotating wrongangle of −60°. As would be appreciated by those of ordinary skill in theart, with the benefit of this disclosure, the amplitude of single cosinewave responses from parallel transmitter-receiver pair (T_(up) and Rx)should be bigger than the amplitude of the single cosine wave responsesfrom the perpendicular transmitter-receiver pair (T_(dn) and Rx). Aswould be appreciated by those of ordinary skill in the art, with thebenefit of this disclosure, this relationship may also be confirmedusing Eq. (16a) and Eq. (16b). As discussed above, in conjunction withFIG. 12, a comparison of the amplitudes of V4 and V5 of Eq. (10) may beused to evaluate which of the two relative azimuthal rotation angles(φ_(t)) from the procedure of the second step reflects the true value ofthe relative azimuthal rotation angle φ_(t). Accordingly, while rotatingthe tool 902 30° to the boundary target, the amplitude of the singlecosine wave of parallel pair increases, whereas that of theperpendicular pair decreases, which verifies that the angle of 30° isthe relative azimuthal angle. FIG. 17 shows signal responses for anexemplary embodiment where the x-axis of the tool coordinate system isparallel to the formation boundary.

EXAMPLE II

A commercially available ADR-TT tool was examined in a water tank with asurrounding conductive casing target to validate the methods disclosedherein. In order to verify the methods disclosed herein, experimentswere conducted in a water tank with a surrounding conductive casingparallel to the tool ADR-TT. In order to steer the casing position, theflowchart in FIG. 12 and the processing schemes described herein wereapplied to the raw measurements from the water tank experiments. FIG. 18represents raw measurements of a spacing of 44 inches betweentransmitter and receiver in FIG. 13 with operating frequency of 500 kHz.FIG. 19 shows the processed signals after rotating the tool high side tothe target casing. Because there is only one target surrounding thetool, it can be proven using Eq. (16a) and Eq. (16b.) that the singlecosine wave responses of perpendicular Tx-Rx pair should be smaller thanthe responses of parallel Tx-Rx pair. Accordingly, the ADR-TT may besteered to the target casing by using the proposed concept andprocessing schemes disclosed herein. The results of the experimentconfirm that the methods disclosed herein may be used to determinesurrounding casing position. Moreover, the experiments confirm that theresponses after the processing scheme are similar to the modelingresponses.

The present invention is therefore well-adapted to carry out the objectsand attain the ends mentioned, as well as those that are inherenttherein. While the invention has been depicted, described and is definedby references to examples of the invention, such a reference does notimply a limitation on the invention, and no such limitation is to beinferred. The invention is capable of considerable modification,alteration and equivalents in form and function, as will occur to thoseordinarily skilled in the art having the benefit of this disclosure. Thedepicted and described examples are not exhaustive of the invention.Consequently, the invention is intended to be limited only by the spiritand scope of the appended claims, giving full cognizance to equivalentsin all respects.

What is claimed is:
 1. A method of processing raw measurements from anelectromagnetic resistivity logging tool comprising: providing anelectromagnetic resistivity logging tool; wherein the electromagneticresistivity logging tool comprises a rotational position sensor; atleast one transmitter antenna; and at least one receiver antenna;obtaining raw measurements at the at least one receiver antenna;averaging all bin measurements at the at least one receiver antenna in afirst step; averaging two raw complex voltage measurements at the atleast one receiver antenna in a second step; wherein the two raw complexvoltage measurements comprise a first raw complex voltage measurement ina bin direction and a second raw complex voltage measurement in anopposite bin direction; and averaging a difference between one rawmeasurement at the at least one receiver antenna in a bin direction fromanother raw measurement at the at least one receiver antenna in oppositebin direction in a third step.
 2. The method of claim 1, furthercomprising distributing the raw measurements at the at least onereceiver antenna into a group consisting of one or more of a constantcomplex voltage; an azimuthal voltage as a sinusoid wave response withdouble periods; and an azimuthal voltage as a sinusoid wave responsewith a single period.
 3. The method of claim 2, further comprising curvefitting the raw measurements obtained at the at least one receiverantenna.
 4. The method of claim 3, wherein curve fitting the rawmeasurements obtained at the at least one receiver antenna comprisesusing cosine curve fitting.
 5. The method of claim 3, further comprisingcalculating signal match with a forwarding model.
 6. The method of claim5, wherein calculating signal match with a forwarding model comprisesdetermining a relative azimuthal rotation angle of the electromagneticresistivity logging tool.
 7. The method of claim 1, further comprisingdetermining a casing position based, at least in part, on the first rawcomplex voltage.
 8. The method of claim 1, wherein the electromagneticresistivity logging tool further comprises a rotational position sensor.9. The method of claim 1, further comprising storing the first rawcomplex voltage.
 10. The method of claim 1, further comprising storingthe raw measurements.
 11. The method of claim 1, further comprising:determining a coupling matrix based, at least in part, on the first rawcomplex voltage; and adjusting a drilling direction based, at least inpart, on the coupling matrix.
 12. A method of processing rawmeasurements from an electromagnetic resistivity logging toolcomprising: providing an electromagnetic resistivity logging tool;wherein the electromagnetic resistivity logging tool comprises: a firsttransmitter antenna oriented in a first quadrant; a receiver antennaoriented in one of the first quadrant and a third quadrant, wherein thethird quadrant is diagonal to the first quadrant; a second transmitteroriented in one of a second quadrant and a fourth quadrant, wherein thesecond quadrant and the fourth quadrant are adjacent to the firstquadrant; wherein a distance between the first transmitter antenna andthe receiver antenna is substantially same as a distance between thesecond transmitter antenna and the receiver antenna; determining anexpression of raw measurements at the receiver antenna in response tofiring of the first transmitter as a first expression; determining anexpression of raw measurements at the receiver antenna in response tofiring of the second transmitter as a second expression; and obtainingan expression for processed signals matching to forward model responsesusing the first expression and the second expression.
 13. The method ofclaim 11, wherein the receiver antenna is substantially parallel to thefirst transmitter antenna.
 14. The method of claim 11, wherein thereceiver antenna is substantially perpendicular to the secondtransmitter antenna.
 15. The method of claim 11, wherein the firstexpression is a function of a tilt angle of the first transmitter, atilt angle of the receiver antenna and a complex voltage matrix.
 16. Themethod of claim 11, wherein the electromagnetic resistivity logging toolfurther comprises a rotational position sensor.
 17. The method of claim11, wherein the electromagnetic resistivity logging tool furthercomprises a rotational position sensor.
 18. The method of claim 11,further comprising storing the raw measurements.
 19. The method of claim11, wherein obtaining an expression for processed signal matching toforward model responses comprises determining a relative azimuthalrotation angle of the electromagnetic resistivity logging tool.
 20. Themethod of claim 1, further comprising: adjusting a drilling directionbased, at least in part, the first expression and the second expression.